Study of the yielding of sand under generalized stress conditions using a versatile hollow cylinder torsional apparatus

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Accepted Manuscript

Brendan C. O’Kelly, Patrick J. Naughton PII: S0167-6636(08)00155-5

DOI: 10.1016/j.mechmat.2008.11.002

Reference: MECMAT 1644

To appear in: Mechanics of Materials Received Date: 15 June 2007

Revised Date: 8 October 2008 Accepted Date: 11 November 2008

Please cite this article as: O’Kelly, B.C., Naughton, P.J., Study of the yielding of sand under generalized stress conditions using a versatile hollow cylinder torsional apparatus, Mechanics of Materials (2008), doi: 10.1016/ j.mechmat.2008.11.002

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Study of the yielding of sand under generalized stress conditions using a

versatile hollow cylinder torsional apparatus

Brendan C. O’Kelly(corresponding author)

Lecturer,

Department of Civil, Structural and Environmental Engineering,

Museum Building, Trinity College Dublin, Dublin 2, Ireland.

Tel. +353 18962387 Fax. +353 16773072 E-mail: bokelly@tcd.ie

Patrick J. Naughton

Lecturer in Civil Engineering,

School of Engineering,

Institute of Technology, Sligo,

Ballinode, Sligo, Ireland.

Tel. +353 719155489 Fax. + 353 719155390 E-mail: naughton.patrick@itsligo.ie

First submission: 15th June 2007

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Abstract

The hollow cylinder torsional apparatus (HCTA) is particularly suited to studying the

mechanical behavior of cross-anisotropic test specimens since the magnitudes of the three

principal stresses and the orientation of the major-minor principal stress axes can be

independently controlled. Novel features of the University College Dublin HCTA,

particularly the instrumentation and calibration procedures necessary to achieve accurate

generalized stress path testing, are presented. The plastic yield behavior of fully saturated

Leighton Buzzard sand was studied under generalized stress conditions. Physically

identical test-specimens (35.5 mm inner radius, 50.0 mm outer radius and 200 mm in

length) were prepared using a water-pluviation technique that reproduces the inherent

cross-anisotropic fabric of many natural sand deposits. A series of stress path tests

systematically probed the stress space to locate segments of yield loci. The deviator stress

at yield was found to vary in a well-defined pattern that depended on the magnitude of the

intermediate principal stress parameter. However, the deviator stress at yield was largely

independent of reorientations of the major principal stress that had occurred during the

initial anisotropic consolidation stage of the tests. The experimental yield data was used to

show that the Matsuoka-Nakai and Lade yield criteria provide satisfactory predictions of

the onset of plastic yielding in sand deposits under generalized stress conditions for routine

geotechnical engineering design.

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Nomenclature

b intermediate principal stress parameter

H specimen length

HCTA hollow cylinder torsional apparatus

J1, J2and J3 first, second and third effective stress invariants

p mean confining stress

p’ mean effective confining stress

piand pe inner and outer confining pressures, respectively

riand re inner and outer specimen radii, respectively

R major-to-minor principal stress ratio

R’ major-to-minor effective principal stress ratio

T torque

q deviator stress

u pore water pressure

v axial displacement

wiand we radial displacements of inner and outer specimen wall surfaces,

respectively

W axial load

 inclination of the major principal stress to the vertical direction

z, r,  axial, radial and circumferential normal strains

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 total normal stress

’ effective normal stress

z, r,  axial, radial and circumferential total normal stresses

1, 2,  major, intermediate and minor total principal stresses

1’, 2’, ’ major, intermediate and minor effective principal stresses

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1. Introduction

Geomaterials behave pseudo-elastically up to about 10-3strain. In elastic-plastic modeling, the transition with increasing stress from pseudo-elastic to elastoplastic behavior occurs at

the yield point. Hence, the estimation of the yield stress is important in engineering design

since it defines the onset of larger plastic deformations with further increases in applied

stress. In most ground engineering problems, the principal stresses rotate as the foundation

load is applied. For example, Fig. 1 shows the stress conditions at five points (labeled A to

E) along a potential rupture surface in the ground. Since most sedimentary deposits are

inherently anisotropic, the stiffness and strength responses of the ground foundation will

vary from point to point along the rupture surface and are functions of the stress history and

both the magnitudes and the directions of the principal stresses. At point D, the major

principal stress (1) has rotated through an angle () to the vertical direction.

Fig. 1. Stress states along rupture surface.

The hollow cylinder torsional apparatus (HCTA) is uniquely capable of independently

controlling the magnitudes of the three principal stresses and the orientation of the

major-minor principal stress axes. Hight et al. (1983) and Saada and Townsend (1981)

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This paper describes the use of a state-of-the-art HCTA developed at University College

Dublin (UCD) to study the plastic yield behavior of a Leighton Buzzard sand under

generalized stress conditions. The development of the apparatus has been reported in detail

by O’Kelly and Naughton (2003, 2005a, 2008a). Additional information is presented in this

paper on its novel features, in particular the instrumentation, automation and calibration

procedures necessary to achieve accurate generalized stress path testing. A series of stress

paths were targeted to locate segments of yield loci for the Leighton Buzzard sand in

generalized stress space.

The experimental yield data was then compared with the values predicted by two existing

yield criteria, namely Matsuoka-Nakai and Lade, in order to assess their accuracy in

predicting the onset of plastic yielding in the sand specimens. The two yield criteria are

particularly suited to soil mechanics and have been shown by Pradel et al. (1990),

Matsuoka et al. (1990) and Lade and Duncan (1976) to provide reasonable predictions for a

range of foundation loading conditions. For example, Matsuoka and Nakai (1985) have

shown that the Matsuokai-Nakai yield criterion had the effect of averaging the Mohr

Coloumb criterion which is widely used in soil mechanics. In contrast, isotropic-kinematic

hardening formulations distinguish between primary and secondary yield whereas the

bounding surface is allowed to shrink and dilate during cyclic loading in models using

bounding surface theory (Pradel et al, 1990). Hence, a direct comparison of the

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difficult to establish and in any event would have had to be examined on a case by case

basis.

The Matsuoka-Nakai yield criterion (Matsuoka and Nakai, 1985), which is a theoretical

development of the concepts of compoundedly mobilized and spatially mobilized planes,

can be expressed in terms of the effective stress invariants, Eq. (1). The Lade yield criterion

(Lade and Duncan, 1975), which was developed from analysis of cubical-triaxial test data

assuming isotropic behavior, can be best expressed in terms of the first and third effective

stress invariants, Eq. (2).

3 2 1

J J J

= Matsuoka-Nakai constant (1)

3 3 1

J J

= Lade constant (2)

3 2 1

1   

J (3a)

1 3 3 2 2 1

2   

J (3b)

3 2 1 3  

J (3c)

where J1, J2and J3are the first, second and third effective stress invariants,

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Previous studies, most notably by Pradel et al. (1990), have used HCTA devices to study

the effects of a reorientation of the major principal stress on the yield behavior of sand, but

only for limited regions of stress space, and with the experimental data reflecting the

combined effects of changes in both and the intermediate principal stress parameter (b):

3 1

3 2

 

  

b = Sin2 (4)

The b parameter is a sizing ratio that relates the magnitude of the intermediate principal

stress to the magnitudes of the major and minor principal stresses (0b 1). In this paper,

the UCD HCTA was used to study the effects of different orientations of the major

principal stress and different magnitudes of the b parameter on the yield behavior of the

sand separately.

2. Principle of hollow cylinder torsional testing

An axial compressive load (W) and a torque (T) are applied across the length of the

thick-walled, hollow-cylindrical test specimen while independently-controlled confining

pressures (piand pe) are applied over the inner and outer specimen wall surfaces, Fig. 2(a).

The normal stresses acting on an element of the ground foundation are invariably

compressive and are simulated in the HCTA by applying the compressive load and

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test specimen (overall length H and inner and outer radii, riand re, respectively) may

deform axially, radially and twist under the applied surface tractions. A polar coordinate

system is appropriate with the z-axis coincident with the specimen axis.

(a) Surface tractions.

(b) Stress components.

[image:10.595.79.495.206.413.2]

(c) Principal stresses.

Fig. 2. Stress state in hollow cylindrical specimen.

Figures 2(b, c) show the non-zero stress components (z, r, , z) and the resolved

principal stresses (1, 2, 3) that are induced in an element of the specimen wall. The

complementary shear stress (z) that acts on the vertical plane automatically develops in

response to the shear stress zinduced by the applied torque. The Lamé equations for

calculating the mean values of the non-zero stress components (Table 1) induced across the

specimen wall thickness were derived from equilibrium considerations assuming an

isotropic, linear-elastic response (Naughton and O’Kelly, 2007). The stress components

were integrated over the full specimen volume in order to account for the effects of the

specimen wall curvature. From numerical analysis of the test specimen, Wijewickreme

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[image:11.595.92.509.196.513.2]

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Table 1. Equations for stress and strain components.

Axis direction Mean stress Mean strain

Axial normal 2 i 2 e 2 i i 2 e e 2 i 2 e z r r r p r p ) r r ( W        H v z  

Radial normal

_

_



₂



2 i 2 e i e i e 2 e 2 i 2 i 2 e 2 i i 2 e e r r r r r ln p p r r 2 r r r p r p              _          i e i e r r r w w Circumferential normal







₂



2 i 2 e i e i e 2 e 2 i 2 i 2 e 2 i i 2 e e r r r r ln p p r r 2 r r r p r p             _ _         _ i e i e r r w w Circumferential

shear 3 (r r )(r r )

) r r ( T 4 2 i 2 e 4 i 4 e 3 i 3 e z       _ ) r r ( H 3 ) r r ( 2 2 i 2 e 3 i 3 e z      _

The stress components (z, , z) are resolved to determine the magnitudes of the major

and minor principal stresses and the orientation () of the major principal stress to the

vertical direction. The intermediate principal stress equals the radial normal stress (r).

Hence, the magnitudes of the three principal stresses and the orientation of the major

principal stress can be independently controlled by varying the applied surface tractions.

The non-zero strain components (z, r, , z) are computed from the measured axial (v)

and twist () deformations and the radial displacements of the inner and outer specimen

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resolved to determine the magnitudes of the three principal strains and the orientation of the

major principal strain to the vertical direction. The HCTA is particularly suited to studying

the constitutive behavior of cross-anisotropic test specimens since the major-minor

principal stress axes can be rotated relative to the cross-anisotropic plane.

Geomaterials generally comprise three phases, namely: the solid particles; a liquid phase,

usually water, and a gaseous phase, usually air (partially saturated conditions). However,

the stiffness and strength responses of the test specimen are customarily measured in the

fully saturated condition in the geotechnical laboratory, in which case the mechanical

response is governed by the effective normal stress () that acts across the contacts

between the constituent solid particles. Hence, the stress state must be expressed in terms of

the effective stress condition in geotechnical analysis:

u

 

 

 (Terzaghi, 1943) (5)

where is the applied or total normal stress;

and u is the pressure in the pore water that fills the saturated voids.

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3.1. Overview of apparatus

The development of the UCD HCTA (Fig. 3) has been reported in detail by O’Kelly and

Naughton (2003, 2005a, 2008a). Additional information on its novel features, particularly

the instrumentation, automation and calibration procedures necessary to achieve accurate

generalized stress path testing, is presented in this paper.

Fig. 3. UCD hollow cylinder torsional apparatus.

The ends of the test specimen are in contact with annular platens inside an acrylic pressure

cell. Hydrostatic confining pressures are applied to the inner and outer specimen wall

surfaces using two screw-driven, pressure actuators (each 2.0 MPa and 200 ml in capacity).

The inner and outer specimen wall surfaces are enclosed by rubber membranes, typically

0.38-mm in thickness. The outer cell chamber and the specimen inner bore cavity are

independently sealed and both chambers are completely filled with water. The rated

capacity of the acrylic cylinder limits the confining pressure that can be applied to the outer

cell chamber to a maximum of 1.0 MPa. The back pressure applied to the pore water in the

specimen and the volume change of the specimen itself (drained test) are controlled by a

third pressure actuator.

An axial load and a torque are applied to the bottom end of the specimen via a loading

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restrained under torsional loading conditions by ribbed, sintered bronze discs that are

secured to both the loading and reaction platens. Rotation of the lightly-preloaded screw

and spline ball bearings, which were secured to the reaction frame located beneath the

pressure cell, can produce smooth vertical, rotary and spiral motions of the loading piston,

and hence the loading platen. The top end of the specimen is restrained by a load reaction

cage which is located inside the acrylic cylinder in the UCD set up. Under quasi-static

loading conditions, the loading mechanisms can apply an axial tension or compression

force of up to 19.3 kN and a torque of up to ±103 N.m.

Practical considerations, including the necessary control sensitivity for the applied surface

tractions, the strain measurement accuracy and the convenient deployment of

instrumentation, particularly in the specimen inner bore cavity, were considered in arriving

at the UCD test specimen dimensions of 35.5-mm inner radius, 50.0-mm outer radius and

200-mm in length. Numerical analysis by Naughton and O’Kelly (2007) indicated that the

stress non-uniformity that may develop across the thickness of the UCD specimen due to its

wall curvature increased in proportion to the major-to-minor principal stress ratio (R),

although the level of stress non-uniformity was generally acceptable throughout the stress

space for R 1.5. The specimen wall thickness of 14.5 mm was also sufficiently large in

relation to the grain size of the sand so that a uniform densification of the specimen could

be achieved during specimen preparation and potential failure mechanisms were free to

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Based on the capacities of the loading mechanisms and pressure actuators, an axial total

normal stress of up to ±5.0 MPa; a radial total normal stress of up to 2.0 MPa and a

circumferential shear stress of up to ±0.6 MPa can be induced in the UCD specimen, which

is more than adequate to study the stress-strain-strength behavior of geomaterials.

Specimens having smaller cross-section areas can be tested by fitting different annular

platens in the pressure cell when higher operating stress levels are required.

Stepper-motors drive the mechanisms that actuate the loading piston and pressure actuators

to apply small axial and twist boundary displacements and confining pressures to the outer

cell chamber and the specimen inner bore cavity. Submersible, high-resolution

instrumentation measures the axial load and torque that develop across the specimen length,

the applied confining pressures and the axial, radial and twist deformations of the

specimen. When the apparatus is operating in precision mode, one step motion by the drive

motors displaces or rotates the loading piston by 1x10-4mm or 0.8 arc-seconds, corresponding to a control resolution of about 7x10-5strain. Errors due to apparatus

compliance and specimen end-restraint and bedding effects (Baldi et al., 1988) are excluded

by measuring the specimen deformations over a central gauge length using instruments

attached or in close proximity to the specimen wall surfaces. The apparatus is located in a

temperature-controlled environment at 202oC.

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3.2.1. Applied load, torque and confining pressures

A combined thrust-torque transducer (4.0 kN and 50 N.m in capacity) measures the axial

and torsional loads that develop across the specimen length. The transducer is located

inside the pressure-cell, next to the restrained end of the specimen, so that measurement

errors due to frictional resistance (for example, along the loading piston) do not arise. The

thrust-torque transducer was electrically insulated inside a saturated, oil-filled housing that

formed part of the reaction platen assembly, Fig. 4.

Fig. 4. Pressure compensation system for thrust-torque transducer.

The transducer readings are unaffected by changes in the confining pressures applied

during a test since the confining pressure to the outer cell chamber also acts on the oil

contained inside the transducer housing, via a water-oil interface. The transducer was

calibrated in-place by applying axial load and torque increments using a system of levers,

pulleys and dead weights. The calibration procedure involved the application of an axial

load alone, torque alone and combined axial and torsional loading, Fig. 5(a). Typical

calibration data for the case of axial loading alone are shown in Fig. 5(b). The coupling

between the axial load and torque signals was analyzed using multiple-linear regression

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Fig. 5. Calibration of thrust-torque transducer.

The two pressure actuators independently control the applied confining pressures to within

0.25 kPa and can target a specified pressure at up to 15 kPa/s. The pressure actuators also

measure the volume change of the outer cell chamber and the specimen inner bore cavity. A

400-kPa differential pressure transducer measures the difference in the confining pressures

acting across the specimen wall-thickness.

3.2.2. Specimen deformations

The polar deformation response of the test specimen is measured locally using inclinometer

gauges and proximity transducers (Figs. 6 and 7) so that the deformation measurements are

free from errors associated with apparatus compliance and specimen end-restraint and

bedding effects (Baldi et al., 1988). The apparatus is supported entirely on pneumatic

mounts which prevent any external vibrations from interfering with the instrument

measurements (particularly important at small strain levels).

Fig. 6. Inclinometer gauges: double-axes (left); single-axis (right).

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The axial and twist specimen deformations are measured over a central 45-mm gauge

length using three inclinometer gauges (O’Kelly and Naughton, 2008a,b) that are glued to

the outer specimen membrane. Two axial and three twist deformation measurements of up

to 20 mm and ±30oare measured over the outer specimen wall surface using one single-axis and two double-axes inclinometer gauges, Fig. 6. The radial displacements of the inner

and outer wall surfaces are measured near the specimen mid-height using two proximity

transducers (O’Kelly and Naughton, 2008c); one transducer is located in the outer cell

chamber and the second is located directly opposite in the specimen inner bore cavity.

A micro-seal waterproofing treatment was applied to the face of the proximity transducers.

Prior to the assembly of the pressure cell, aluminum foil targets are attached by a smear of

silicone grease to the rubber membranes that enclose the specimen wall surfaces,

preventing any specimen reinforcement effect. Tatsuoka et al. (1983) reported that the

displacement of the rubber membranes, and hence the foil targets, matched that of the

specimen wall surfaces. Each foil target is made large enough to accommodate the

anticipated relative movements between the proximity transducers and the specimen wall

surfaces during the course of a test.

The inclinometer gauges and proximity transducers were calibrated using an optical table

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The proximity transducers were calibrated while fully submerged under water since the

transducer response is sensitive to its operating environment. The gap between the

transducer face and the targets (two plastic discs covered with aluminum foil that had the

same curvatures as the inner and outer specimen wall surfaces) were measured using a laser

distance-measurement system that had resolution and accuracy values of the order of 10-12 and 10-11m, respectively. The data curves for calibration in air and fully submerged in water were found to differ by between 0.07 and 0.09 mm over the measurement range of

0.5 to 5.0 mm. The local instrumentation had a measurement range of between 10-5and 10-1 strain (i.e. pseudo-elastic to failure strain levels for geomaterials), covering the full strain

range of engineering interest. Existing HCTA devices have had to incorporate dynamic

measurement techniques, including shear wave velocity and resonant column tests

(Connolly and Kuwano, 1999; Drnevich, 1985), in order to extend the strain measurement

capability into the pseudo-elastic range. The output signals from the different instruments

were recorded at 5 min intervals over the 24 hour period proceeding each stress path test

and the electrical stability of the outputs was found to be satisfactory (standard deviation

from the mean output voltage was generally significantly less than 0.5%). Longer-term

checks were periodically carried out and again the stability of the outputs was found to be

satisfactory.

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Figure 8 shows a schematic of the data acquisition and control systems for the UCD

HCTA. The drive units and the pressure actuators interface with a personal computer via

Compumotorinterconnect cards and a general-purpose interface bus. The apparatus was

closed-loop controlled to target a user-defined stress path using a suite of LabVIEW

[image:20.595.81.499.201.440.2]

programs, Fig. 9.

Fig. 8. Schematic of data acquisition and control systems.

Fig. 9. Flow chart for control algorithms.

Firstly, the prescribed stress path is read from a data file. The load, pressure and

deformation instruments are simultaneously scanned to determine their initial reference

outputs. The target values of the surface tractions are computed from the mean effective

stress components that correspond to a series of transition stress points located along the

stress path. The loading mechanisms and pressure actuators apply small axial and twist

displacements and confining pressures to the specimen surfaces by independently

controlling the direction and step rate of the motor drives. The axial load and torque that are

induced across the specimen length are controlled by the thrust-torque transducer in a

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and the specimen deformations. The actual state of effective stress and the strain response

are computed in real time using the equations in Table 1. The two axial and three twist

deformations measured by the inclinometer gauges are used to compute the specimen axial

normal and circumferential shear strain values, respectively. The radial displacements of

the specimen wall surfaces recorded by the proximity transducers are used to calculate the

radial and circumferential normal strain values. The stress path is followed by sequentially

stepping between transition target stress-points located along the prescribed stress path.

The stiffness and strength of sand and gravel deposits (frictional materials) are directly

related to the major-to-minor effective principal stress ratio, R’. Hence, it is preferable to

express the stress path in terms of an alternative but equivalent set of control parameters,

namely: the mean effective confining stress (p’); b parameter; R’ and (Table 2).

Table 2. Control parameters used in the algorithms.

Stress parameter Equation Max difference

between

target stress points

Mean effective confining

stress 3

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Intermediate principal stress

parameter 1 3

3 2 ' ' ' ' b        0.1 Major-to-minor effective

principal stress ratio 3

1 ' ' '    R 0.1 Major-to-minor principal stress difference

t = '1–'3 5.0 kPa

Angle of major principal

stress to vertical direction

                ' ' 2 Tan 2 1 z z 1 5.0 degree

The key change points along the stress path, as well as the transition target stress points, are

expressed in terms of the set of control parameters. The control algorithm loops and the

loading mechanisms and pressure actuators are simultaneously adjusted to bring all of the

effective stress components within specified tolerances for the target values (default set at

0.25 kPa). The algorithm considers the impact that a change in the value of one control

parameter would have on the other parameters in determining step size taken by the drive

motors. The specimen deformations are allowed to equilibrate at each target point along the

stress path. Equilibrium is deemed to have been achieved when the rate of change in the

strain components are all within the limits specified in Table 3. The equilibrium check is

bypassed if the specimen has not achieved equilibrium within a specified number of loops

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prescribed stress path as closely as possible. The control algorithm includes calibration

equations for the different instruments and automatically corrects the specimen stresses for

the restraining effects of the rubber membranes that enclose the outer and inner specimen

[image:23.595.76.493.238.600.2]

wall surfaces (Sivathayalan and Vaid, 1998; Tatsuoka et al., 1986).

Table 3. Limiting rates that identify whether the specimen has achieved equilibrium.

Stress component Unit Rate

Axial, radial and circumferential

normal strains strain/min

5x10–5

Circumferential shear strain 1x10–5

Pore water pressure kPa/min 0.5

4. Study of the plastic yield behavior of sand

4.1. Overview

The UCD HCTA was used to study the plastic yield behavior of a Leighton Buzzard sand

under generalized stress conditions, and in particular the effects of a reorientation of the

major principal stress and changes in the magnitude of the b parameter. The stress paths

systematically probed generalized stress space using the methodology of Tatsuoka and

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developing yield loci to be identified for the sand. After isotropically consolidating the test

specimens to the same mean effective confining stress, new fabrics were induced in the

specimens by following prescribed anisotropic consolidation stages that targeted different

values, and with the intermediate principal stress parameter b = 0 or 0.5. Yield points

were then identified for prescribed b parameter values in the range 0 ≤ b ≤ 1.0by increasing

the R’ value until plastic straining commenced. This systematic approach allowed the

effects of the reorientation of the major principal stress and the changes in the magnitude of

the b parameter to be independently studied. The accuracy of the Matsuoka-Nakai and Lade

yield criteria (Eqs. (1) and (2)) in predicting the onset of plastic yielding in the sand

specimens was assessed by comparing the predicted and experimental yield values.

4.2. Materials and specimen preparation method

The test material was a white, clean, fine-to-medium Leighton Buzzard sand that comprised

sub-angular grains, about 0.5-mm in size (Table 4). The sand in its loosest state had a

maximum void ratio (defined as the volume of the void space to the volume of the sand

grains) of 0.80. In its densest state, the sand had a minimum void ratio of 0.53. Eight,

physically identical test-specimens were prepared using a water-pluviation technique

(O’Kelly and Naughton, 2005b), which mimicked the sedimentation process, producing

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were set up on the specimen loading platen. The test specimens were prepared to an initial

target void ratio of 0.73 by slowly depositing the saturated sand into the annular,

water-filled cavity between the cylindrical moulds. The eight test specimens were densified to a

medium dense state (target void ratio of 0.60) by uniformly tapping around the outer mould

using a mallet. The local instrumentation was set up after removing the preparation moulds.

Next, the pressure cell was fitted and secured to the support frame. The outer cell chamber

and the specimen inner bore cavity were simultaneously flooded with water and

pressurizing to 150 kPa, and with a backpressure of 100 kPa applied to the specimen (p’ =

[image:25.595.78.439.304.644.2]

50 kPa). Full saturation of the sand was achieved in this manner.

Table 4. Physical properties of Leighton Buzzard sand.

Coefficient of uniformity, CU 1.32

Coefficient of curvature, CZ 0.96

Effective grain size, D10(mm) 0.44

Specific gravity of solids 2.64

Maximum void ratio 0.80

Minimum void ratio 0.53

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The stress path tests comprised four stages, as follows, with Fig. 10 showing a

characteristic stress path plotted in terms of R’ versus the b parameter:

Stage 1: The inner and outer confining pressures were simultaneously increased from 150

to 200 kPa and the specimen was allowed to drain overnight against the applied

specimen back pressure which was continuously adjusted in order to maintain the

mean effective confining stress p’ = 100 kPa (point A).

Stage 2: Next the specimen was consolidated anisotropically by increasing the effective

principal stress ratio R’ from 1.0 to 1.5 and re-orientating the major principal

stress to an angle of = 0, 30, 60 or 90 degrees from the vertical direction,

which induced a in the specimens. During this stage, the b parameter value was

either maintained constant at b = 0 or increased from b = 0 to 0.5. The values of

p’ and were then held constant along the remaining stages of the stress path.

Stage 3: A yield point (point B) was established by increasing the R’ value until plastic

straining of the specimen commenced, which was characterized by a marked

change in its stiffness response. An initial yield point was established at R’ = 2.1

for the eight specimens tested in this study.

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With the b parameter value set, the R’ value was increased again to search for a

second yield point (point E). This allowed a yield locus to be established between

points B and E. The procedure was repeated to establish segments of yield loci for

[image:27.595.309.495.213.355.2]

different stress states in the specimens.

Fig. 10. Stress probing to identify yield loci.

The test program identified 16 segments of yield loci distributed throughout the stress

space. Care was taken to limit the development of plastic straining during the

reconsolidation stages of the stress paths. Significant plastic straining would have increased

the transition zone between pseudo-elastic and post-yield behavior making it more difficult

to accurately define the yield point value.

4.4. Experimental results and analysis

The eight test-specimens had been prepared to similar initial void ratio values of between

0.59 and 0.61 using a wet-pluviation technique, and hence had similar cross-anisotropic

fabrics initially. Naughton and O’Kelly (2003) have shown experimentally that the

stress-strain response of the Leighton Buzzard sand was not significantly affected by this level of

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A typical data set, including the actual and target stress paths and the measured strain

responses, is shown in Fig. 11 for the case of p’ = 100 kPa,= 30o, and with the b parameter value being increased in 0.25 steps from b = 0.5 at the end of the anisotropic

consolidation stage 2 to b = 1.0 by the end of the stress path test. The prescribed stress path

is defined by the continuous line segments in Fig. 11(a). The key change points on the

stress path and the corresponding strain values are also labeled alphabetically, namely

points A to K. Figure 11(b) shows the manner in which the four stress components (z’, r’,

’, z) were continuously adjusted by the control program in order to follow the

prescribed stress path to within 2% of the target values. Figures 11(c, d) show the strain

responses recorded during the load-unload and reconsolidation stages of the stress path (i.e.

changes in the R’ and b parameter values, respectively). The stress-strain responses in Fig.

11(c) were approximately linear-elastic along the unload paths and along the initial phase

of the reload paths. Furthermore, the strains during the reconsolidation stages were

negligible. Hence, the soil fabric induced in the specimens by the end of the anisotropic

consolidation stage 2 would have remained largely intact for the remainder of the stress

path.

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However, the stress-strain responses became increasingly non-linear on nearing a yield

point. Pradel et al. (1990) suggested that the yield points derived from the intersection of

the best-fit lines to the initial pseudo-elastic and post-yield slopes provided the most

appropriate estimate of the yield loci for the Matsuokai-Nakai and Lade yield criteria.

Hence, the yield point was identified by curve-fitting the experimental plots. A segment of

the yield locus was defined by joining the yield point established for higher values of the b

parameter back to the initial yield point. Minor discrepancies arose in the experimental

values determined for a specific yield point identified by curve-fitting analysis of the

different stress-strain responses (coefficient of variation in the R’ value of between 0.2%

and 10.9%). Similar differences were also reported in the experimental study by Pradel et

al. (1990). Hence, the stress state corresponding to a specific yield point in this study was

calculated as the mean value from curve-fitting analysis of a minimum of four experimental

stress-strain curves.

Figure 12 shows the experimental yield data plotted in terms of , the b parameter and the

deviator stress (q), given by:



 

_'



2

3 ' 2 2 ' 2 ' 1 2 ' 3 '

1     

     



q (6)

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The experimental data suggests that the magnitude of the deviator stress corresponding to

the yield loci varies continuously, although in a well-defined pattern, namely: the deviator

stress at yield was dependent on the initial value of the b parameter but was largely

independent of the rotation that had occurred during the anisotropic consolidation stage

of the stress path. It is plausible that the densification of the specimens by tapping around

the cylindrical moulds during specimen preparation had significantly reduced the initial

level of cross-anisotropy that had been created by the wet-pluviation technique. Two trends

are observed with respect to the b parameter, namely:

(i) Stress paths that had re-orientated the major principal stress, and with the

intermediate principal stress parameter value initially set at b = 0: the deviator stress

at yield decreased by typically 15% to 20% due to the increase in the b parameter

value from b = 0 to 0.5;

(ii) Stress paths that had re-orientated the major principal stress, and with the b

parameter value initially set at 0.5: the magnitude of the deviator stress at yield

either remained constant or increased marginally due to the increase in the b

parameter value from to b = 0.5 to 1.0.

The experimental yield loci are compared with the predictions given by the

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the respective yield constants (Eqs. (1) and (2)), and normalized to allow direct

comparisons with the theoretical values predicted by the two yield criteria.

(a) Matsuoka-Nakai criterion.

(b) Lade criterion.

Fig. 13. Experimental and theoretical values of points on yield loci.

The two yield criteria were found to slightly underestimate the onset of yielding for 0b

0.5 and slightly overestimate the onset of yielding for 0.5< b 1.0. However, given the

approximate nature of the curve-fitting analysis that had been used in determining the

experimental yield values, it is concluded that both yield criteria provide satisfactory

predictions of the onset of plastic yielding in sand under generalized stress conditions. The

margin of error in the experimental yield loci from the curve-fitting analysis (coefficient of

variation in the R’ value of between 0.2% and 10.9%) prohibits the identification of the

specific mechanism(s) that give rise to the minor discrepancies with the two yield criteria

studied since several mechanisms may be simultaneously mobilized. The experimental

yield loci may also resemble inflated balloons in principal stress space rather than the

conical shapes with smooth triangular cross-sections given by the theoretical models.

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intermediate principal stress parameter in the prediction of the onset of plastic yielding in

sand deposits for routine geotechnical engineering design.

5. Summary and conclusions

A versatile hollow cylinder torsional apparatus (HCTA) was used to systematically probe

generalized stress space in order to locate segments of yield loci for a fully saturated

Leighton Buzzard sand. Eight, physically identical test-specimens (35.5-mm inner radius,

50.0-mm outer radius and 200-mm in length) were prepared using a water-pluviation

technique that mimicked the sedimentation process, and hence the inherent

cross-anisotropic fabric of many natural sand deposits. The stresses induced in the test specimens

were continually adjusted to within 2% of the target values on the prescribed stress paths,

which comprised in sequence: an isotropic consolidation stage to the same mean effective

confining stress; an anisotropic consolidation stage that produced different reorientations of

the major principal stress and finally probing of the stress space to locate segments of

developing yield loci. The specimen deformation response was measured to a resolution of

10-5strain over its central gauge length. Novel features of the HCTA, particularly the instrumentation, automation and calibration procedures that were necessary to achieve

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on the initial value of the intermediate principal stress parameter but was largely

independent of the stress reorientation that occurred during the anisotropic consolidation

stage of the stress path. The Matsuoka-Nakai and Lade yield criteria were both found to

provide satisfactory predictions of the onset of plastic yielding in the sand under

generalized stress conditions for routine geotechnical engineering design.

Acknowledgements

The authors would like to acknowledge the support of Dr. Tom Widdis and Professor

Eugene O’Brien during the course of the research, which was carried out at the Department

of Civil Engineering, University College Dublin (UCD). The paper was written by the lead

author while on sabbatical leave at the Urban Institute Ireland, UCD. The lead author is

also grateful for funding received through the Pierse-Newman Scholarship in Civil

Engineering from UCD.

References

Baldi, G., Hight, D.W., Thomas, G.E., 1988. A reevaluation of conventional triaxial test

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Connolly, T.M., Kuwano, R., 1999. The measurement of Gmaxin a resonant column, bender

elements, torsional shear apparatus. In: Proceedings Second International Symposium on

Pre-failure Deformation Characteristics of Geomaterials, Torino, Italy, 1, 73–80.

Drnevich, V.P., 1985. Recent developments in resonant column testing. In: Proceedings

Richart Commemorative Lectures, ASTM Specialty Session, Detroit, pp. 79–107.

Hight, D.W., Gens, A., Symes, M.J., 1983. The development of a new hollow cylinder

apparatus for investigating the effects of principal stress rotation in soils. Géotechnique

33(4), 355–383.

Lade, P.V., Duncan, J.M., 1975. Elastoplastic stress strain theory for cohesionless soil.

ASCE Geotechnical Engineering J. 101(GT10), 1037–1053.

Matsuoka, H., Nakai, T., 1985. Relationship among Tresca, Mises, Mohr-Coulomb and

Matsuoka-Nakai failure criteria. Soils and Foundations 25(4),123–128.

Matsuoka, H., Suzuki, Y., Murata, T., 1990. A constitutive model for soils evaluating

principal stress rotation and its application to some deformation problems. Soils and

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Naughton, P.J., O’Kelly, B.C., 2003. The anisotropy of Leighton Buzzard sand under

general stress conditions. In: Proceedings Third International Symposium on Deformation

Characteristics of Geomaterials, Lyon, France, 1, 285–291.

Naughton, P.J., O'Kelly, B.C., 2007. Stress non-uniformity in a hollow cylinder torsional

sand specimen. Geomechanics and Geoengineering 2(2), 117–122.

O’Kelly, B.C., Naughton, P.J., 2003. Development of the University College Dublin

hollow cylinder apparatus. In: Proceedings 56th Canadian Geotechnical Conference,

Winnipeg, Ontario.

O’Kelly, B.C., Naughton, P.J., 2005a. Development of a new hollow cylinder apparatus for

stress path measurements over a wide strain range. ASTM Geotechnical Testing J. 28(4),

345–354.

O’Kelly, B.C., Naughton, P.J., 2005b. Engineering properties of wet-pluviated hollow

cylindrical specimens. ASTM Geotechnical Testing J. 28(6), 570–576.

O'Kelly B.C.; Naughton P.J., 2008a. Local measurements of the polar deformation response

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O’Kelly B.C.; Naughton P.J., 2008b. Measuring shear strain in a hollow cylinder apparatus

using electrolevel-inclinometer gages. In: Proceedings Fourth International Symposium on

Pre-Failure Deformation Characteristics of Geomaterials, Atlanta, Georgia, USA.

O’Kelly B.C.; Naughton P.J., 2008c. Use of proximity transducers for local radial strain

measurements in a hollow cylinder apparatus. In: Proceedings Fourth International

Symposium on Pre-Failure Deformation Characteristics of Geomaterials, Atlanta, Georgia,

USA.

Pradel, D., Ishihara, K., Gutierrez, M., 1990. Yielding and flow of sand under principal

stress axes rotation. Soils and Foundations 30(1), 87–99.

Saada, A.S., Townsend, F.C., 1981. State of the art: laboratory strength testing of soils. In:

Laboratory Shear Strength of Soils, ASTM STP740, Philadelphia, pp. 7–77.

Sivathayalan, S., Vaid, Y.P., 1998. Truly undrained response of granular soils with no

membrane penetration effects. Canadian Geotechnical J. 35, 730–739.

Tatsuoka, F., Ishihara, K., 1974. Yielding of sand in triaxial compression. Soils and

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Tatsuoka, F., Muramatsu, M., Saski, T., 1983.Closure toCyclic undrained stress-strain

behavior of dense sands by torsional simple shear tests. Soils and Foundations 23(3), 142–

145.

Tatsuoka, F., Sonoda, M., Hara, K., Fukushima, S., Pradhan, T.B.S., 1986. Failure and

deformation of sand in torsional shear. Soils and Foundations 26(4), 79–97.

Terzaghi, K., 1943. Theoretical Soil Mechanics. John Wiley and Sons, New York.

Wijewickreme, D., 1990. Behavior of sand under simultaneous increase in stress ratio and

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Figure captions

(a) Surface tractions.

(b) Stress components.

(c) Principal stresses.

Fig. 2. Stress state in hollow cylinder torsional specimen.

[image:38.595.75.501.167.726.2]

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Fig. 7. Specimen set up inside the pressure cell.

Fig. 8. Schematic of data acquisition and control systems.

Fig. 9. Flow chart for control algorithms.

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Fig. 12. Variation of deviator stress on yield loci.

(b) Lade criterion.

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Table captions

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Study of the yielding of sand under generalized stress conditions using a

versatile hollow cylinder torsional apparatus

[image:42.595.72.503.194.722.2]

Brendan C. O’Kelly and Patrick J. Naughton

(b) Stress components. (c) Principal stresses.

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[image:43.595.79.493.99.610.2]

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[image:43.595.94.402.384.713.2]

Inclinometers Proximity transducer Thrust-torque transducer

Test specimen

Stepper motors

Gearbox

Screw-spline shaft Pulleys and tensioned cables

Displacement transducer Rotary encoder Support frame Acrylic cylinder

Loading piston Loading platen Reaction platen

Thrust-torque transducer

Reaction platen

Test specimen Oil-filled cavity

Oil inlet

Specimen inner

bore cavity Specimen top cap

Air breed Transducer

housing

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[image:44.595.215.498.98.367.2]

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[image:44.595.97.376.393.617.2]

Fig. 5. Calibration of thrust-torque transducer.

25 mm

(a) Loading sequences.

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[image:45.595.102.502.97.663.2]

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Fig. 7. Specimen set up inside the pressure cell.

Thrust-torque

transducer

Reaction platen

Proximity transducer Test specimen Annular, porous disc with protruding ribs

Specimen inner

bore cavity

Loading platen Locating dowels

Loading piston Outer

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[image:46.595.171.499.67.355.2]

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[image:46.595.96.393.380.699.2]

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[image:47.595.101.503.90.747.2]

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BE and FK are segments of developing yield loci

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(a)

(b)

(c)

[image:48.595.94.501.72.739.2]

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[image:49.595.95.494.63.714.2]

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Fig. 12. Variation of deviator stress on yield loci.

0.8 0.9 1 1.1 1.2

0 0.25 0.5 0.75 1

Intermediate principal stress parameter, b

M a ts u o k a -N a k a i N o rm a liz e d C o n s ta n t

 = 300

 = 600

 = 00

 = 900

0.8 0.9 1 1.1 1.2

0 0.25 0.5 0.75 1

Intermediate principal stress parameter, b

L a d e N o rm a liz e d C o n s ta n

t  = 300

 = 600

 = 00

 = 900

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Study of the yielding of sand under generalized stress conditions using a

versatile hollow cylinder torsional apparatus

[image:50.595.80.515.224.699.2]

Brendan C. O’Kelly and Patrick J. Naughton

Axis direction Mean stress Mean strain

Axial normal 2 i 2 e 2 i i 2 e e 2 i 2 e z r r r p r p ) r r ( W      

 z _Hv

Radial normal

_

_



₂



2

i 2 e i e i e 2 e 2 i 2 i 2 e 2 i i 2 e e r r r r r ln p p r r 2 r r r p r p                        i e i e r r r w w Circumferential

normal







₂



2

[image:50.595.82.517.521.725.2]

i 2 e i e i e 2 e 2 i 2 i 2 e 2 i i 2 e e r r r r ln p p r r 2 r r r p r p             _          _ i e i e r r w w Circumferential shear ) r r )( r r ( 3 ) r r ( T 4 2 i 2 e 4 i 4 e 3 i 3 e z       _ ) r r ( H 3 ) r r ( 2 2 i 2 e 3 i 3 e z      _

Stress parameter Equation Max difference between

target stress points

Mean effective confining

stress p 1 ₃2 3

     

 5.0 kPa

Intermediate principal stress parameter 3 1 3 2 ' ' ' ' b        0.1 Major-to-minor effective principal stress ratio

3 1 ' ' '    R 0.1 Major-to-minor principal stress difference

t = '1–'3 5.0 kPa

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